Theorem sotri2 6127

Description: A transitivity relation. (Read 𝐴 ≤ 𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

Assertion

Ref	Expression
sotri2	⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5724	. . . 4 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
3	2	simpld 500	. . 3 ⊢ (𝐵𝑅𝐶 → 𝐵 ∈ 𝑆)
4		soi.1	. . . . . . 7 ⊢ 𝑅 Or 𝑆
5		sotric 5597	. . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵)))
6	4, 5	mpan 703	. . . . . 6 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵)))
7	6	con2bid 357	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8		breq1 5110	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 ↔ 𝐴𝑅𝐶))
9	8	biimpd 232	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
10	4, 1	sotri 6125	. . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
11	10	ex 418	. . . . . 6 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
12	9, 11	jaoi 871	. . . . 5 ⊢ ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
13	7, 12	biimtrrdi 257	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)))
14	13	com3r 88	. . 3 ⊢ (𝐵𝑅𝐶 → ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)))
15	3, 14	mpand 708	. 2 ⊢ (𝐵𝑅𝐶 → (𝐴 ∈ 𝑆 → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)))
16	15	3imp231 1130	1 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902   class class class wbr 5107   Or wor 5566   × cxp 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-po 5567  df-so 5568  df-xp 5665

This theorem is used by:  supsrlem  11123